Olympiad problems

2021 Regional Competition For Advanced Students, 2

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

2011 Sharygin Geometry Olympiad, 22

Tags: tangent , geometry , euler circle , circumcircle , concurrency

Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.

1994 All-Russian Olympiad, 3

Tags: geometry , tangent

Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at A and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that the common chord of the circumcircles of triangles $ABC$ and $BDE$ passes through point $F$. (A. Kalinin)

2016 Saudi Arabia IMO TST, 3

Tags: geometry , circles , tangent

Given two circles $(O_1)$ and $(O_2)$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $(O_1)$ and $(O_2)$ at $G, E$ ($\ne A$), respectively, the line $d_2$ cuts $(O_1)$ and $(O_2)$ at $F, H$ ($\ne A$), respectively, such that $E$ is between $A, G$ and $F$ is between $A, H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $(O_1), (O_2)$ at $K, L$ ($\ne B$), respectively. Let $N$ be the intersection of $O_1K$ and $O_2L$. Prove that the circle $(NLK)$ is tangent to $AB$.

2008 Estonia Team Selection Test, 5

Tags: midpoint , tangent , geometry , circles

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

2002 Abels Math Contest (Norwegian MO), 3a

Tags: tangent , geometry , circles

A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.

2025 Junior Macedonian Mathematical Olympiad, 5

Tags: geometry , circumcircle , ratio , tangent

Let $M$ be the midpoint of side $BC$ in $\triangle ABC$, and $P \neq B$ is such that the quadrilateral $ABMP$ is cyclic and the circumcircle of $\triangle BPC$ is tangent to the line $AB$. If $E$ is the second common point of the line $BP$ and the circumcircle of $\triangle ABC$, determine the ratio $BE: BP$.

2015 Costa Rica - Final Round, 6

Tags: geometry , reflection , tangent , equal angles

Let $\vartriangle ABC$ be a triangle with circumcenter $O$. Let $ P$ and $Q$ be internal points on the sides $AB$ and $AC$ respectively such that $\angle POB = \angle ABC$ and $\angle QOC = \angle ACB$. Show that the reflection of line $BC$ over line $PQ$ is tangent to the circumcircle of triangle $\vartriangle APQ$.

2009 Belarus Team Selection Test, 1

Tags: circles , equal segments , tangent , tangent circle , geometry

Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$. I. Voronovich

2015 Belarus Team Selection Test, 2

Tags: geometry , tangent , circles , median

The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$. I. Gorodnin

2017 Ukrainian Geometry Olympiad, 3

Tags: geometry , circumcircle , tangent , right triangle

On the hypotenuse $AB$ of a right triangle $ABC$, we denote a point $K$ such that $BK = BC$. Let $P$ be a point on the perpendicular from the point $K$ to line $CK$, equidistant from the points $K$ and $B$. Let $L$ be the midpoint of $CK$. Prove that line $AP$ is tangent to the circumcircle of $\Delta BLP$.

2018 Argentina National Olympiad, 6

Tags: geometry , tangent , parallelogram , circles

Let $ABCD$ be a parallelogram. An interior circle of the $ABCD$ is tangent to the lines $AB$ and $AD$ and intersects the diagonal $BD$ at $E$ and $F$. Prove that exists a circle that passes through $E$ and $F$ and is tangent to the lines $CB$ and $CD$.

2009 Regional Olympiad of Mexico Northeast, 3

Tags: incircle , tangent , circumcircle , geometry

The incircle of triangle $\vartriangle ABC$ is tangent to side $AB$ at point $P$ and to side $BC$ at point $Q$. The circle passing through points $A,P,Q$ intersects line $BC$ a second time at $M$ and the circle passes through the points $C,P,Q$ and cuts the line $AB$ a second time at point$ N$. Prove that $NM$ is tangent to the incircle of $ABC$.

2011 Oral Moscow Geometry Olympiad, 1

Tags: geometry , circumcircle , tangent , symmetry

$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: tangent , geometry , equal angles , circles

Two circles $C_1(O_1)$ and $C_2(O_2)$ with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.

2021 South Africa National Olympiad, 4

Tags: circles , geometry , tangent , circumcircle

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$. Let $K$ be the circle through $A$ with centre $B$. Let $D$ be the other intersection of $K$ and $AC$. Let $K$ intersect the circumcircle of $BCD$ again at $E$. If $F$ is the intersection of $DE$ and $BH$, show that $BD$ is tangent to the circle through $D$, $F$, and $H$.

2009 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: geometry , isosceles , tangent , circumcircle

Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove: $a)$ Triangle $AMD$ is isosceles $b)$ Circumcenter of $AMD$ lies on circle $C$

2018 Junior Regional Olympiad - FBH, 4

Tags: circles , touching , tangent , radius

It is given $4$ circles in a plane and every one of them touches the other three as shown: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc=[/img] Biggest circle has radius $2$, and every one of the medium has $1$. Find out the radius of fourth circle.

1969 IMO Shortlist, 47

Tags: geometry , circumcircle , reflection , tangent

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

2017 IMO Shortlist, G2

Tags: geometry , circumcircle , tangent

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

2009 IMO Shortlist, 4

Tags: cyclic quadrilateral , projective geometry , tangent , power of a point , geometry

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2017 Peru IMO TST, 3

Tags: geometry , circumcircle , incircle , midpoint , tangent

The inscribed circle of the triangle $ABC$ is tangent to the sides $BC, AC$ and $AB$ at points $D, E$ and $F$, respectively. Let $M$ be the midpoint of $EF$. The circle circumscribed around the triangle $DMF$ intersects line $AB$ at $L$, the circle circumscribed around the triangle $DME$ intersects the line $AC$ at $K$. Prove that the circle circumscribed around the triangle $AKL$ is tangent to the line $BC$.

2022 Taiwan TST Round 2, 5

Tags: geometry , circumcircle , miquel point , complete quadrilateral , tangent

Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$. A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively. Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$. Prove that, if the line $AD$ is tangent to the circle $\odot(DXY)$, then the line $AE$ is tangent to the circle $\odot(EXY)$. [i]Proposed by ltf0501.[/i]

2015 Junior Balkan Team Selection Tests - Moldova, 3

Tags: tangent , geometry , circumcircle , parallel

Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.

2003 Junior Tuymaada Olympiad, 7

Tags: tangent , geometry , angle bisector , perpendicular

Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.